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Maths assignment XII
RELATIONS AND FUNCTIONS
5
−7
1. If 𝑓(𝑥) = [𝑥] and (𝑥) = |𝑥| , then evaluate (𝑓𝑜𝑔) (2) − (𝑔𝑜𝑓) ( 2 ).
1
2. If 𝑓(𝑥) = (3 − 𝑥 3 )3 and (𝑥) = log 𝑒 𝑥 , find 𝑓𝑜𝑓(𝑥).
3. Let * be a binary operation defined by a*b = 2ab – 7. Is * associative?
4. Let 𝐴 = {1,2,3} and = {4,5,6} , 𝑓: 𝐴 → 𝐵 is a function defined on 𝑓(1) = 4 , 𝑓(2) =
5 and 𝑓(3) = 6. Write the inverse of f as a set of ordered pairs.
5. Let ‘*’ be a binary operation defined on the set Z as a a*b = a+b+1 for a,b ∈ I. find
the identity element.
𝑥−1
6. If 𝑓(𝑥) =
; 𝑥 ≠ −1 , then find 𝑓 −1 (𝑥).
𝑥+1
7. Show that the signum function given by 𝑓(𝑥) = 1, if x is greater than 0, if x=0, -1 , if
x is less than 0 is neither one-to-one nor onto.
8. If 𝑓: 𝑅 → 𝑅, a,b,c,d ∈ R such that ( a,b ) * ( c,d ) = (ac, b+ad ). Find the identity
element.
9. Give an example for a relation which is neither reflexive nor symmetric.
10. Consider 𝑓, 𝑔 ∶ 𝑁 → 𝑁 and ℎ ∶ 𝑁 → 𝑅 defined as 𝑓(𝑥) = 2𝑥 , 𝑔(𝑦) = 3𝑦 + 4 ,
and ℎ(𝑧) = 𝑠𝑖𝑛𝑧 for all x,y,z ∈ N. Show that ho(gof) = (hog)of.
11. Let A = R X R and let * be a binary operation on A defined by ( a,b )*( c,d ) = ( ad+bc ,
bd ) for all ( a,b ) , ( c,d ) ∈ R X R. Determine if * is commutative , associative , has
identity and inverse.
12. Let A = {-1 , 0 , 1 , 2 } , B = [ -4 , -2 , 0 , 2 } and 𝑓, 𝑔 ∶ 𝐴 → 𝐵 be the function defined by
1
𝑓(𝑥) = 𝑥 2 − 𝑥, 𝑥 ∈ 𝐴 and 𝑔(𝑥) = 2 |𝑥 − 2| − 1 , 𝑥 ∈ 𝐴 ; are f and g equal ? Justify
your answer.
5𝑥+3
13. If 𝑓(𝑥) = 4𝑥−5, show that f(f(x)) is an identity function.
14. Let 𝑓(𝑥) = [𝑥] and 𝑔(𝑥) = |𝑥|. Find
−5
−5
(i) (𝑔𝑜𝑓) ( ) − (𝑓𝑜𝑔) ( )
3
5
5
3
(ii) (𝑔𝑜𝑓) (3) − (𝑓𝑜𝑔) (3)
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
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(iii) (𝑓 + 2𝑔)(−1)
15. Check whether the relation R defined in the set {1,2,3,4,5,6} as R = { (a,b) : b = a+1 }
is reflexive, symmetric or transitive.
16. Show that the relation S in the set given by S = { (a,b : |a-b| is even } is an
equivalence relation. Also find the set of all elements related to 4.
1
17. If f(x) = (3 − 𝑥 3 )3 , find f(f(x)).
18. If 𝑓, 𝑔: 𝑅 → 𝑅 are defined respectively by 𝑓(𝑥) = 𝑥 2 + 3𝑥 + 1 , 𝑔(𝑥) = 2𝑥 − 3. Find
(i) fog (ii) gof (iii) fof (iv) gog.
19. If the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 2𝑥 3 + 7, prove that f is one-one and
onto function. Also find the inverse of the function 𝑓 and 𝑓 −1 (23).
20. If P(X) is the power set of the set X, and A*B=A∪B, find the identity if it exists.
𝑎+𝑏
𝑎+𝑏
21. Examine if the following are binary operations (i) a*b = 2 , a,b ∈ N (ii) a*b = 2 ,
a,b ∈ Q.
22. Consider 𝑓: 𝑅 → [5, ∞) given by 𝑓(𝑥) = 9𝑥 2 + 6𝑥 − 5. Show that f is invertible and
(𝑦+6)−1
𝑓 −1 (𝑦) = √
3
.
23. Let A = N X N and R be the relation on A defined by ( a,b ) R ( c,d ) => (𝑎𝑑(𝑏 + 𝑐) =
𝑏𝑐(𝑎 + 𝑑)). Show that R is an equivalence relation.
24. Let A = N X N and * be the binary operation on A defined by ( a,b )*( c,d ) = ( a+c,b+d
). Show that * is commutative and associative. Find the identity element for * on A, if
any.
−3
2𝑥
25. Let 𝑓: 𝑅 − { 5 } → 𝑅 be a function define as 𝑓(𝑥) = 5𝑥+3 , find 𝑓 −1 : range of f: 𝑅 −
−3
{ 5 }.
26. Consider 𝑓: {1,2,3} → {𝑎, 𝑏, 𝑐} and 𝑔: (𝑎, 𝑏, 𝑐} → {𝑎𝑝𝑝𝑙𝑒, 𝑏𝑎𝑙𝑙, 𝑐𝑎𝑡} defined by f(1)=
a, f(2) =b . f(3)=c , g(a)= apple, g(b)=ball , g(c)=cat. Show that f,g and gof are
invertible. Find out 𝑓 −1 , 𝑔−1 and (𝑔𝑜𝑓)−1 and show that (𝑔𝑜𝑓)−1 = 𝑓 −1 o 𝑔−1 .
27. Prove that the function 𝑓: 𝑅 → 𝑅 defined as 𝑓(𝑥) = 2𝑥 − 3 is invertible and 𝑓 −1 (𝑥).
28. Show that the relation are in the set a = { x : x ∈ w , x≤ 10 } given by R = { (a,b) : |ab| is a multiple of 3 } is an equivalence relation , find the elements related to 3.
INVERSE TRIGONOMETRIC FUNCTIONS
1
5 √2
1. Find 2𝑡𝑎𝑛−1 ( ) + 𝑠𝑒𝑐 −1 (
5
2. Solve for x :
𝑥−1
7
1
) + 2𝑡𝑎𝑛−1 (8)
𝑥+1
(i) 𝑡𝑎𝑛−1 (
) + 𝑡𝑎𝑛−1 (𝑥+2) =
𝑥−2
1−𝑥
1
𝑡𝑎𝑛−1 ( ) = 𝑡𝑎𝑛−1 𝑥
1+𝑥
2
−1 (𝑐𝑜𝑠𝑥)
−1
𝜋
4
(ii)
(iii) 2𝑡𝑎𝑛
= 𝑡𝑎𝑛 (2𝑐𝑜𝑠𝑒𝑐𝑥)
𝜋
−1
−1
(iv) 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 2𝑥 =
(v) 𝑠𝑖𝑛
−1 (1
(vi)𝑐𝑜𝑡
−1
3
−1
− 𝑥) − 2𝑠𝑖𝑛 𝑥 =
𝑥 − 𝑐𝑜𝑡
−1 (𝑥
+ 2) =
𝜋
2
𝜋
12
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
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(vii) 𝑡𝑎𝑛−1 (
2𝑥+1
1−4 𝑥
−1 2
)
(viii) 𝑠𝑖𝑛 (𝑥 √1 − 𝑥 2 − 𝑥√1 − 𝑥 4
3. Simplify :
1
(i) 𝑡𝑎𝑛−1 ( 𝑠𝑖𝑛−1 (
2𝑥
1
1−𝑦 3
)) + 2 𝑐𝑜𝑠 −1 (1+𝑦2 )
1+𝑥 2
2
𝑥−𝑥 −1
(ii) 𝑐𝑜𝑠 −1 ( −1)
𝑥+𝑥
−1
(iii) 𝑡𝑎𝑛 (√1 + 𝑥 2
− 𝑥)
4. Prove that:
12
4
63
(i) 𝑠𝑖𝑛−1 ( ) + 𝑐𝑜𝑠 −1 ( ) + 𝑡𝑎𝑛−1 ( ) = 𝜋
13
5
16
(ii) 𝑐𝑜𝑠 −1 𝑥 + 𝑐𝑜𝑠 −1 𝑦 + 𝑐𝑜𝑠 −1 𝑧 = 𝜋 , prove that 𝑥 2 + 𝑦 2 + 𝑧 2 +
2𝑥𝑦𝑧 = 1
𝑥
𝑥2
𝑦
(iii) 𝑐𝑜𝑠 −1 ( ) + 𝑐𝑜𝑠 −1 ( )=𝛼, prove that 2 −
𝑎
𝑏
𝑎
1
2
1
3
(iv) 𝑡𝑎𝑛−1 ( ) + 𝑡𝑎𝑛−1 ( ) = 𝑐𝑜𝑠 −1 ( )
4
9
2
5
4
5
16
2𝑥𝑦
𝑎𝑏
𝑐𝑜𝑠𝛼 +
𝑦2
𝑏2
= 𝑠𝑖𝑛2 𝛼
𝜋
(v) 𝑠𝑖𝑛−1 ( ) + 𝑠𝑖𝑛−1 ( ) + 𝑠𝑖𝑛−1 ( ) =
5
13
65
2
2 (𝑡𝑎𝑛−1
2 (𝑐𝑜𝑡 −1
(vi) 𝑠𝑒𝑐
2) + 𝑐𝑜𝑠𝑒𝑐
3) = 15
−1
−1
−1
(vii) 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦 + 𝑡𝑎𝑛 𝑧 = 𝜋, prove that 𝑥 + 𝑦 + 𝑧 = 𝑥𝑦𝑧
MATRICES
1. If A = [𝑎𝑖𝑗 ] , where [𝑎𝑖𝑗 ] = {
𝑖 + 𝑗 , 𝑖𝑓 𝑖 ≥ 𝑗
, construct a 3 X 2 matrix A.
𝑖 − 𝑗 𝑖𝑓 𝑖 < 𝑗
3 −2
) , then find k if 𝐴2 = kA – 2I.
4 −2
Construct a 3 X 2 matrix whose elements in the ith row and jth column are given by
𝑖+4𝑗
𝑎𝑖𝑗 = 2 .
2 3
If f(x) = 𝑥 2 − 4𝑥 + 1 , find f(A) , when A = [
].
1 2
1 −2
5 4
Find the matrix X , for which [
] X=[
].
1 3 𝑛
1 1
𝑏(𝑎 −1)
𝑛
𝑎 𝑏
If A = [
] , 𝑎 ≠ 1 , prove that 𝐴𝑛 = [ 𝑎
𝑎−1 ] , n∈N .
0 1
0
1
3 1
For the matrix A = [
] , find a and b such that 𝐴2 + 𝑎𝐼 = 𝑏𝑎 , where I is a 2 X 2
7 5
identity matrix .
0 −1
1 −1
Find X and Y , given that 3X - Y = [
] and X - 3Y = [
].
0 −1
−1 1
3 1
If A = [
] , show that 𝐴2 -5A + 7I = 0 , hence find 𝐴−1 .
−1 2
2. If A = (
3.
4.
5.
6.
7.
8.
9.
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
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−1
10. If A = [ 2 ] , B = [-2 -1 -4] , verify that (AB)’= B’A’ .
3
11. If A = (
cos 𝜃
𝑖 sin 𝜃
𝑖 sin 𝜃
) , then prove by principle of Mathematical Induction that
cos 𝜃
cos 𝑛𝜃 𝑖 sin 𝑛𝜃
𝐴𝑛 = [
].
𝑖 sin 𝑛𝜃 cos 𝑛𝜃
12. Express the following matrix as the sum of a symmetric and a skew-symmetric matrix
1 3 5
: [−6 8 3] .
−4 6 5
2 4
5 4
13. If A = [
] , B =[
] , then verify that (AB)’= B’A’.
3 5
3 2
3 2 5
14. Let A = [4 1 3] . Express A as a sum of two matrices such that one is symmetric
0 6 7
and the other is skew-symmetric.
15. Using elementary transformations , find the inverse of the following matrix :
1
2
3
[2
5
7 ].
−2 −4 −5
3 6
5 2
16. Find the matrices X and Y , if X + Y =[
] and X – Y = [
] and hence find 𝑋 2 −
0 −1
0 9
𝑌2 .
17. Show that the matrix B’AB is symmetric or skew symmetric according as A is
symmetric or skew symmetric.
DETERMINANTS
𝑎2 + 2𝑎
1. Using properties of determinants, prove that : | 2𝑎 + 1
3
𝑎2 2𝑎𝑏 𝑏 2
2. Show that : | 𝑏 2
𝑎2 2𝑎𝑏 | = (𝑎3 + 𝑏 3 )2 .
2𝑎𝑏 𝑏 2
𝑎2
0 1 1
3. Find 𝐴−1 , if 𝐴 = [1 0 1]. Also show that 𝐴−1 =
1 1 0
𝐴2 −3𝐼
2
2𝑎 + 1 1
𝑎 + 2 1| = (𝑎 − 1)3 .
3
1
.
4. Using properties of determinants, prove that :
𝑏+𝑐
|𝑐 + 𝑎
𝑎+𝑏
𝑐+𝑎
𝑎+𝑏
𝑏+𝑐
𝑎+𝑏
𝑏 + 𝑐 | = 2(𝑎 + 𝑏 + 𝑐)(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 − 𝑎2 − 𝑏 2 − 𝑐 2 ).
𝑐+𝑎
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
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𝑠𝑖𝑛(𝐴 + 𝐵 + 𝐶)
−𝑠𝑖𝑛𝐵
5. If 𝐴 + 𝐵 + 𝐶 = 𝜋 , show that [
cos(𝐴 + 𝐵)
𝑠𝑖𝑛𝐵
0
−𝑡𝑎𝑛𝐴
𝑎+𝑏
6. Prove using the properties of determinants : | 𝑏 + 𝑐
𝑐+𝑎
.
𝑐𝑜𝑠𝐶
𝑡𝑎𝑛𝐴] = 0 .
0
𝑏+𝑐
𝑐+𝑎
𝑎+𝑏
𝑐+𝑎
𝑎
|
=
2
|
𝑎+𝑏
𝑏
𝑏+𝑐
𝑐
𝑏
𝑐
𝑎
𝑐
𝑎|
𝑏
7. Using the properties of determinants prove that :
(𝑏 + 𝑐)2
| 𝑏2
𝑐2
𝑎2
(𝑐 + 𝑎)2
𝑐2
𝑎2 +𝑏 2
𝑐
𝑐
8. Show that :
𝑎2
𝑏 2 | = 2𝑎𝑏𝑐 (𝑎 + 𝑏 + 𝑐)3 .
(𝑎 + 𝑏)2
|
| 𝑎
𝑐
|
𝑎 |
𝑎2 +𝑏 2
𝑎
𝑐 2 +𝑎2
𝑏
𝑏
𝑏
9. Using the properties of determinants , show that :
𝑎
|𝑎 + 2𝑏
𝑎+𝑏
𝑎+𝑏
𝑎
𝑎 + 2𝑏
𝑎 + 2𝑏
𝑎 + 𝑏 | = 9𝑏 2 (𝑎 + 𝑏).
𝑎
2
3
10
𝑥
𝑦
𝑧
10. Solve for x,y,z : − +
4
6
5
6
9
20
𝑥
𝑦
𝑧
𝑥
𝑦
𝑧
=4; − + =1; + −
=2.
11. Using the properties of determinants , prove the following :
𝑎
|𝑎 − 𝑏
𝑏+𝑐
𝑏
𝑏−𝑐
𝑐+𝑎
12. Solve for x , |
13. If A = [
𝑐
𝑐 − 𝑎 | = 𝑎3 + 𝑏 3 + 𝑐 3 − 3𝑎𝑏𝑐.
𝑎+𝑏
3𝑥 − 8
3
3
1
−𝑡𝑎𝑛𝑥
3
3𝑥 − 8
3
3
3 | = 0.
3𝑥 − 8
𝑡𝑎𝑛𝑥
𝑐𝑜𝑠2𝑥
] , show that 𝐴′ 𝐴−1 = [
1
𝑠𝑖𝑛2𝑥
−𝑠𝑖𝑛2𝑥
].
𝑐𝑜𝑠2𝑥
14. Using the properties of determinants, prove that :
−𝑏𝑐
2
(i) | 𝑎 + 𝑎𝑐
𝑎2 + 𝑎𝑏
𝑏 2 + 𝑏𝑐
−𝑎𝑐
𝑏 2 + 𝑎𝑏
𝑐 2 + 𝑏𝑐
𝑐 2 + 𝑎𝑐| = (𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎)3 .
−𝑎𝑏
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Page 5
𝑎
(ii) | 𝑏
𝑎𝑥 + 𝑏𝑦
𝑏
𝑐
𝑏𝑥 + 𝑐𝑦
𝑏+𝑐
15. Show that : |𝑐 + 𝑎
𝑎+𝑏
𝑎𝑥 + 𝑏𝑦
𝑏𝑥 + 𝑐𝑦 | = (𝑏 2 − 𝑎𝑐)(𝑎𝑥 2 + 2𝑏𝑥𝑦 + 𝑐𝑦 2 ).
0
𝑎−𝑏
𝑏−𝑐
𝑐−𝑎
𝑎
𝑏 | = 3𝑎𝑏𝑐 − 𝑎3 − 𝑏 3 − 𝑐 3 .
𝑐
CONTINUITY AND DIFFERENTIABILITY
1. Discuss the continuity :
𝑥
,𝑥 ≠ 0
2|𝑥|
(a) 𝑓(𝑥) = { 1
,𝑥 = 0
2
3
2
1
−𝑥,
3
(b) 𝑓(𝑥) =
2
3
≤𝑥<1
2
,𝑥 = 1
+𝑥 ,𝑥 > 1
{
2
𝑐𝑜𝑠3𝑥−𝑐𝑜𝑠𝑥
(c) 𝑓(𝑥) = {
,𝑥 ≠ 0
𝑥2
−4 , 𝑥 = 0
, 𝑥≠1
𝑥−1
[𝑥]−1
(d) 𝑓(𝑥) = {
−1 , 𝑥 = 1
2. Find the value of k such that the function ‘f’ is continuous :
𝑘𝑐𝑜𝑠𝑥
𝜋
,𝑥 ≠ 2
𝜋−2𝑥
(a) 𝑓(𝑥) = {
𝜋
3,𝑥 = 2
1−𝑠𝑖𝑛3 𝑥
3𝑐𝑜𝑠2 𝑥
,𝑥 <
𝑎 ,𝑥 =
(b) 𝑓(𝑥) =
𝑏(1−𝑠𝑖𝑛𝑥)
{
(𝜋−2𝑥)2
1−𝑐𝑜𝑠4𝑥
𝜋
𝜋
2
2
,𝑥 >
𝜋
2
,𝑥 < 0
𝑎 ,𝑥 = 0
𝑥2
(c) 𝑓(𝑥) =
√𝑥
{√16+√𝑥−4
,𝑥 > 0
sin(𝑎+1)𝑥+𝑠𝑖𝑛𝑥
𝑥
𝑐 ,𝑥 = 0
(d) 𝑓(𝑥) =
{
√𝑥+𝑏𝑥 2 −√𝑥
𝑥−5
𝑥−|5|
(e) 𝑓(𝑥) =
,𝑥 < 0
𝑏√𝑥 3
,𝑥 > 0
+ 𝑎 ,𝑥 < 5
𝑎 + 𝑏 ,𝑥 = 5
𝑥−5
{𝑥−|5| + 𝑏 , 𝑥 > 5
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Page 6
𝑥 2 −3𝑥+2
(f) 𝑓(𝑥) = {
𝑥 2 −1
2
3
,𝑥 ≠ 1
𝑘 ,𝑥 = 1
1. If y=sin−1 (𝑥√1 − 𝑥 − √𝑥 √1 − 𝑥 2 )
or
sin−1(𝑥 2 √1 − 𝑥 2 − 𝑥√1 − 𝑥 4 )
𝑦
𝑑𝑦
𝑥+𝑦
2. If log(𝑥 2 + 𝑦 2 ) = 2 tan−1 (𝑥 ) , show that 𝑑𝑥 = 𝑥−𝑦
3. If 𝑥 = 𝑎(𝑐𝑜𝑠𝜃 + 𝜃𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(𝑠𝑖𝑛𝜃 − 𝜃𝑐𝑜𝑠𝜃), find
𝑠𝑖𝑛2 (𝑎+𝑦)
𝑑𝑦
𝑑2 𝑦
𝑑𝑥 2
4. If 𝑠𝑖𝑛𝑦 = 𝑥𝑠𝑖𝑛(𝑎 + 𝑦), prove that 𝑑𝑥 = 𝑠𝑖𝑛𝑎
5. Differentiate 𝑥 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 𝑡𝑎𝑛𝑥
𝑑𝑦
−𝑦
−1
−1
6. 𝑥 = √𝑎sin 𝑡 , 𝑦 = √𝑎cos 𝑡 , prove that =
𝑑𝑥
𝑥
7. 𝑦 = 3 cos(𝑙𝑜𝑔𝑥) + 4 sin(𝑙𝑜𝑔𝑥), prove that 𝑥 2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0
−1
8. 𝑦 = 𝑒 𝑚 sin 𝑥 , prove that (1 − 𝑥 2 )𝑦2 − 𝑥𝑦1 − 𝑚2 𝑦 = 0
3𝑥+4√1−𝑥 2
9. 𝑦 = cos−1 (
5
𝑑𝑦
) , find 𝑑𝑥
2𝑥+1
𝑑𝑦
10. 𝑦 = cos−1 (1+4𝑥 ) , find 𝑑𝑥
𝑑𝑦
11. 𝑦 = log10 (𝑙𝑜𝑔𝑠𝑖𝑛𝑥) , find 𝑑𝑥
𝜋
𝜋
12. 𝑦 = log tan ( + ), show that
13. 𝑓(𝑥) =
4
2
−1 1−𝑥
tan (1+𝑥)
𝑑𝑦
𝑑𝑥
𝑥+2
= 𝑠𝑒𝑐𝑥
− tan−1 (1−2𝑥) , find 𝑓 ′ (𝑥)
𝑡
𝑑2 𝑦
14. 𝑥 = 𝑎(𝑐𝑜𝑠𝑡 + log 𝑡𝑎𝑛 2 ) , 𝑦 = 𝑎(1 + 𝑠𝑖𝑛𝑡), find 𝑑𝑥 2
15. 𝑦 = (cot −1 𝑥)2 , prove that (𝑥 2 + 1)2 𝑦2 + (2𝑥)(𝑥 2 + 1)𝑦1 − 2 = 0
1. If 𝑥 = 𝑠𝑖𝑛𝑡 , 𝑦 = 𝑠𝑖𝑛𝑝𝑡 , prove that (1 − 𝑥 2 )𝑦2 𝑥. 𝑦1 + 𝑝2 𝑦 = 0
𝑑𝑦
𝑙𝑜𝑔𝑥
2. 𝑥 𝑦 = 𝑒 𝑥−𝑦 , prove that 𝑑𝑥 = (1+𝑙𝑜𝑔𝑥)2
𝑑𝑦
−1
3. 𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0 , prove that 𝑑𝑥 = (1+𝑥)2
4. √1 − 𝑥 6 + √1 − 𝑦 6 = 𝑎(𝑥 3 − 𝑦 3 ), prove that
𝑑𝑦
𝑑𝑦
𝑑𝑥
=
𝑥 2 √1−𝑦6
𝑦 2 √1−𝑥 6
𝑦
5. 𝑥 𝑝 𝑦 𝑞 = (𝑥 + 𝑦)𝑝+𝑞 , prove that 𝑑𝑥 = 𝑥
𝑑2 𝑦
6. 𝑦 = 𝑐𝑜𝑡𝑥 + 𝑐𝑜𝑠𝑒𝑐𝑥, prove that prove that (1 − 𝑐𝑜𝑠𝑥)2 𝑑𝑥 2 = 𝑠𝑖𝑛𝑥
1−𝑥
𝑑𝑦
−𝑥
7. If 𝑦 = 𝑠𝑖𝑛 (2 tan−1 (√1+𝑥)) ,prove that 𝑑𝑥 = √1−𝑥 2
√1+𝑥 2 −√1−𝑥 2
8. Differentiate tan−1 (√1+𝑥 2
9. 𝑦 =
𝑥.𝑠𝑖𝑛4 𝑥
√1−𝑥 2
+√1−𝑥 2
) w.r.t tan−1(√1 + 𝑥 2 − 𝑥)
𝑑𝑦
+ 𝑙𝑜𝑔(√1 − 𝑥 2 ) prove that 𝑑𝑥 =
sin−1 𝑥
3
(1−𝑥 2 )2
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Page 7
..
∞
𝑑𝑦
10. 𝑦 = 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥 , find 𝑑𝑥
2
11. 𝑦 = (log(𝑥 + √𝑥 2 + 1)) prove that (𝑥 2 + 1)𝑦2 + 𝑥𝑦1 − 2 = 0
12. If 𝑦 = 𝑒 𝑎𝑥 𝑠𝑖𝑛𝑏𝑥, prove that 𝑦2 − 2𝑎𝑦1 + (𝑎2 + 𝑏 2 )𝑦 = 0
13. If 𝑦 = sin(𝑚 sin−1 𝑥), prove that prove that (1 − 𝑥 2 )𝑦2 − 𝑥𝑦1 + 𝑚2 𝑦 = 0
1
14. If 𝑥 = sin (𝑎 log 𝑦) , prove that (1 − 𝑥 2 )𝑦2 − 𝑥𝑦1 + 𝑎2 𝑦 = 0
15. Find the points of discontinuity , 𝑓(𝑥) = |𝑥| − |𝑥 + 1|
16. Verify mean value theorem , 𝑓(𝑥) = 𝑠𝑖𝑛4 𝑥 + 𝑐𝑜𝑠 4 𝑥
𝑓(𝑥) = (𝑥 2 − 4𝑥 + 3)𝑒 2𝑥
17. Verify Rolle’s theorem , 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 2)2
𝑓(𝑥) = 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥
𝜋
[0, 2 ]
[1,3]
[1,2]
[0,2𝜋]
VECTORS
1. If 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0, show that 𝑎⃗ × 𝑏⃗⃗ = 𝑏⃗⃗ × 𝑐⃗ = 𝑐⃗ × 𝑎⃗.
2. If 𝑎⃗ × 𝑏⃗⃗ = 𝑐⃗ × 𝑑⃗, 𝑎⃗ × 𝑐⃗ = 𝑏⃗⃗ × 𝑑⃗, prove that (𝑎⃗ − 𝑑⃗) ∥ (𝑏⃗⃗ − 𝑐⃗) ,
𝑎⃗ ≠ 𝑑 , 𝑏⃗⃗ ≠ 𝑐⃗.
3. Prove that |𝑎⃗ × 𝑏⃗⃗|2 = |𝑎⃗|2 |𝑏⃗⃗|2 − (𝑎⃗. 𝑏⃗⃗)2 , |𝑎⃗| = 5 , |𝑏⃗⃗| = 3, |𝑎⃗ × 𝑏⃗⃗| =
25, find (𝑎⃗. 𝑏⃗⃗).
4. The dot product of a vector with vectors 𝑖̂ + 𝑗̂ − 3𝑘̂ , 𝑖̂ + 3𝑗̂ − 2𝑘̂ ,
2𝑖̂ + 𝑗̂ + 4𝑘̂ are 0 , 5 and 8. Find the vector.
5. Show that 2𝑖̂ − 𝑗̂ + 𝑘̂ , 𝑖̂ − 3𝑗̂ − 5𝑘̂ and 3𝑖̂ − 4𝑗̂ − 4𝑘̂ form the sides of a
right angled triangle.
6. Find the unit vector perpendicular to the plane ABC , position vectors of
A , B and C are 2𝑖̂ − 𝑗̂ + 𝑘̂ , 𝑖̂ + 𝑗̂ + 2𝑘̂ and 2𝑖̂ + 3𝑘̂ respectively.
7. Express 𝑎⃗ = 5𝑖̂ − 2𝑗̂ + 5𝑘̂ as the sum of two vectors such that one is
⃗⃗⃗⃗
parallel to the vector 𝑏⃗⃗ = 3𝑖̂ + 𝑘̂ and the other is perpendicular to 𝑏.
8. If 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 3𝑘̂ , 𝑏⃗⃗ = 3𝑖̂ − 𝑗̂ + 2𝑘̂ , show that
(i) (𝑎⃗ + 𝑏⃗⃗) ⊥ (𝑎⃗ − 𝑏⃗⃗)
(ii) Find a unit vector perpendicular to both (𝑎⃗ + 𝑏⃗⃗) and (𝑎⃗ − 𝑏⃗⃗).
9. Find the projection of 𝑏⃗⃗ + 𝑐⃗ on 𝑎⃗ , where 𝑎⃗ = 2𝑖̂ − 2𝑗̂ + 𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ −
2𝑘̂ and 𝑐⃗ = 2𝑖̂ − 𝑗̂ + 4𝑘̂.
10.If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑏⃗⃗ = 𝑗̂ − 𝑘̂ , find a vector 𝑐⃗ such that 𝑎⃗ × 𝑐⃗ = 𝑏⃗⃗ , 𝑎⃗. 𝑐⃗ =
3.
11.If 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0 , |𝑎⃗| = 3 , |𝑏⃗⃗| = 5 , |𝑐⃗| = 7 , find the angle between
𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗.
12.If 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗ and 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗ , 𝑎⃗ ≠ 0, prove that 𝑏⃗⃗ = 𝑐⃗.
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Page 8
13.If 𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ are position vectors of A , B and C of ∆𝐴𝐵𝐶, show that
1
(i) area of ∆𝐴𝐵𝐶 = |𝑎⃗ × 𝑏⃗⃗ + 𝑏⃗⃗ × 𝑐⃗ + 𝑐⃗ × 𝑎⃗|
2
(ii) deduce the condition of collinear.
14.If 𝑎⃗ × 𝑏⃗⃗ = 𝑏⃗⃗ × 𝑐⃗, show that 𝑎⃗ + 𝑐⃗ = 𝑚𝑏⃗⃗ , where m is a scalar.
15.If 𝑎⃗ is a unit vector , (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗) = 8 , find |𝑥|.
16.Find the area of ∆𝐴𝐵𝐶 , A(1,1,1) , 𝐵(1,2,3), 𝐶(2,3,1).
17.If |𝑎⃗| = 3 , |𝑏⃗⃗| = 4 , |𝑐⃗| = 5 and one of them is perpendicular to the sum
of other two, find |𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗|.
⃗⃗⃗⃗, 𝑐⃗ are mutually
18.If If 𝑎⃗ × 𝑏⃗⃗ = 𝑐⃗ , 𝑏⃗⃗ × 𝑐⃗ = 𝑎⃗ , prove that 𝑎⃗ , 𝑏
perpendicular, |𝑏⃗⃗| = 1 , |𝑐⃗| = |𝑎⃗|.
𝜃
1
𝜃
1
19.If 𝑎̂ , 𝑏̂ are unit vectors , prove that sin = |𝑎⃗ − 𝑏⃗⃗| , cos = |𝑎⃗ + 𝑏⃗⃗ |.
2
2
2
2
20.If 𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ are of equal magnitude and mutually perpendicular , show
that 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ is equally inclined to 𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ .
𝜋
21.𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ are unit vectors 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗ = 0 , angle between 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ is .
6
Prove that 𝑎⃗ = ±2(𝑏⃗⃗ × 𝑐⃗).
22.If the sum of two unit vectors is a unit vector , prove that magnitude of
their difference is √3.
23.Find the area of the parallelogram whose diagonals are 3𝑖̂ + 𝑗̂ − 2𝑘̂ and
𝑖̂ − 3𝑗̂ + 4𝑘̂.
24. Prove that 𝑎⃗ × (𝑏⃗⃗ + 𝑐⃗) + 𝑏⃗⃗ × (𝑐⃗ + 𝑎⃗) + 𝑐⃗ × (𝑎⃗ + 𝑏⃗⃗) = ⃗0⃗.
*************
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Page 9
Rubrics
Criteria being evaluated
Criteria being evaluated
Task is
complete
Task is
incomplete
Answers is
complete
Answer is not
complete with
some
confusion
Subject / Grade / Work Sheet Number / Chapter Number / 2012-2013
Task is not
completed
Answer is not
correct.
student needs
teachers
support to
complete the
task
Page 10